We study the problem of edge partitioning, where the goal is to partition the edge set of a graph into several parts. The replication factor of a vertex $v$ is the number of parts that contain edges incident to $v$. The goal is to minimize the average replication factor of the vertices while keeping the sizes of the parts nearly equal. We study the regime where the number of parts is significantly smaller than the size of the graph. To this end, we introduce a new class of edge partitioning algorithms. These algorithms guarantee asymptotically worst-case-optimal upper bounds on the replication factor for any constant number of parts $k$, and when $k$ grows slowly with the number of vertices. In particular, we show that the optimal replication factor for growing $k$ is $\sqrt{k}(1+o(1))$. The algorithms are computationally efficient, including in the LOCAL and CONGEST models, and can be implemented as stateless streaming algorithms in graph processing frameworks. Some of the worst-case graphs are complete graphs and jumbled graphs, also known as pseudo-random graphs. Our method generalizes a family of algorithms based on symmetric intersecting families of sets. Informally, we replace the symmetry condition by a weaker balance condition that is still sufficient for the algorithms. This relaxation makes it possible to construct such families with asymptotically optimal rank $\sqrt{k}(1+o(1))$.

翻译：我们研究边划分问题，目标是将图的边集划分为若干部分。顶点$v$的副本因子指包含该顶点关联边的部分数量。目标是在保持各部分规模近似相等的同时，最小化顶点平均副本因子。我们研究部分数量显著小于图规模的场景。为此，我们提出一类新型边划分算法。对于任意常数$k$及随顶点数缓慢增长的$k$，这些算法保证副本因子的渐近最坏情况最优上界。特别地，我们证明当$k$增长时，最优副本因子为$\sqrt{k}(1+o(1))$。算法计算高效，适用于LOCAL和CONGEST模型，并可在图处理框架中实现为无状态流算法。部分最坏情况图包括完全图和伪随机图（亦称混杂图）。我们的方法推广了基于对称相交族的一类算法。非正式地说，我们用更弱的平衡条件替代对称性条件，该条件对算法仍然充分。这一松弛使得构造渐近最优秩为$\sqrt{k}(1+o(1))$的这类族成为可能。